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Equilibrium Statistical Physics Physics Course Materials

2015

10. Canonical Ensemble IGerhard MüllerUniversity of Rhode Island, gmuller@uri.edu

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AbstractPart ten of course materials for Statistical Physics I: PHY525, taught by Gerhard Müller at theUniversity of Rhode Island. Documents will be updated periodically as more entries becomepresentable.

This Course Material is brought to you for free and open access by the Physics Course Materials at DigitalCommons@URI. It has been accepted forinclusion in Equilibrium Statistical Physics by an authorized administrator of DigitalCommons@URI. For more information, please contactdigitalcommons@etal.uri.edu.

Recommended CitationMüller, Gerhard, "10. Canonical Ensemble I" (2015). Equilibrium Statistical Physics. Paper 5.http://digitalcommons.uri.edu/equilibrium_statistical_physics/5

Contents of this Document [ttc10]

10. Canonical Ensemble I

• Canonical ensemble. [tln51]

• Classical ideal gas (canonical ensemble). [tex76]

• Ultrarelativistic classical ideal gas (canonical idela gas). [tex77]

• Ultrarelativistic classical ideal gas in two dimensions. [tex154]

• Array of classical harmonic oscillators (canonical ensemble). [tex78]

• Irreversible decompression. [tex136]

• Irreversible heat exchange. [tex137]

• Reversible decompression. [tex139]

• Reversible heat exchange. [tex140]

• Heavy piston. [tex141]

• Ensemble averages. [tln52]

• Classical virial theorem. [tln83]

• Systems of noninteracting particles. [tln54]

• Further ensemble averages. [tln55]

• Classical ideal gas in a uniform gravitational field. [tex79]

• Gas pressure and density inside centrifuge. [tex135]

• Relative momentum of two ideal gas particles. [tex80]

• Partition function and density of states. [tln56]

• Ideal gas partition function and density of states. [tex81]

• Vibrational heat capacities of solids. [tln57]

• Array of quantum harmonic oscillators (canonical ensemble). [tex82]

• Vibrational heat capacities of solids (Debye theory). [tsl29]

• Thermodynamic perturbation expansion. [tln80]

• Vibrational heat capacity of a solid. [tex83]

• Anharmonic oscillator and thermodynamic perturbation. [tex104]

Canonical Ensemble [tln51]

Consider a closed classical system (volume V , N particles, temperature T ).The goal is to determine the thermodynamic potential A(T, V, N) pertainingto that situation, from which all other thermodynamic properties can bederived.

Maximize Gibbs entropy S = −kB

∫Γ

d6NX ρ(X) ln[CNρ(X)]

subject to the constraints related to normalization and average energy:∫Γ

d6NX ρ(X) = 1,

∫Γ

d6NX H(X)ρ(X) = U.

Apply calculus of variation with two Lagrange multipliers:

δ

∫Γ

d6NX−kBρ ln[CNρ] + α0ρ + αUHρ = 0

⇒∫

Γ

d6NX δρ−kB ln[CNρ]− kB + α0 + αUH = 0.

⇒ · · · = 0 ⇒ ρ(X) =1

CN

exp

(α0

kB

− 1 +αU

kB

H(X)

).

Determine the Lagrange multipliers α0 and αU :∫Γ

d6NX ρ(X) = 1 ⇒ exp

(1− α0

kB

)=

1

CN

∫Γ

d6NX exp

(αU

kB

H(X)

)≡ ZN .

∫Γ

d6NX ρ(X)· · · = 0 ⇒ S − kB + α0 + αUU = 0.

⇒ U +1

αU

S =kB

αU

ln ZN . Compare with U − TS = A ⇒ αU = − 1

T.

Helmholtz free energy: A(T, V, N) = −kBT ln ZN .

Canonical partition function: ZN =1

CN

∫Γ

d6NX exp (−βH(X)) , β =1

kBT.

Probability density: ρ(X) =1

ZNCN

exp (−βH(X)) .

Canonical ensemble in quantum mechanics:

ZN = Tre−βH =∑

λ

e−βEλ , ρ =1

ZN

e−βH , A = −kBT ln ZN .

[tex76] Classical ideal gas (canonical ensemble)

Consider a classical ideal gas of N atoms confined to a box of volume V in thermal equilibriumwith a heat reservoir at temperature T . The Hamiltonian of the system reflects the kinetic energyof 3N noninteracting degrees of freedom:

H =3N∑i=1

p2i

2m.

(a) Show that the canonical partition function is ZN = V N/(N !λ3NT ), where λT =

√h2/2πmkBT

is the thermal wavelength.(b) Derive from ZN the Helmholtz free energy A(T, V,N), the entropy S(T, V,N), the pressurep(T, V,N), the internal energy U(T,N), and the chemical potential µ(T, V ).(c) Show that the pressure is equal to two thirds of the energy density and that the adiabatessatisfy p3V 5 = const.

Solution:

[tex77] Ultrarelativistic classical ideal gas (canonical ensemble)

Consider a classical ideal gas of N atoms confined to a box of volume V in thermal equilibriumwith a heat reservoir at an extremely high temperature T . The Hamiltonian of the system,

H =N∑

l=1

|pl|c,

where c is the speed of light, reflects the ultrarelativistic energy of N noninteracting particles:(a) Calculate the canonical partition function ZN of this system.(b) Derive from ZN the Helmholtz free energy A(T, V,N), the entropy S(T, V,N), the pressurep(T, V,N), the internal energy U(T,N), and the chemical potential µ(T, V ).(c) Show that the pressure is equal to one third of the energy density and that the adiabates satisfyp3V 4 = const.

Solution:

[tex154] Ultrarelativistic classical ideal gas in two dimensions

Consider a classical ideal gas of N particles confined to a two-dimensional box of area V in thermalequilibrium at extremely high temperature T . Most particles are moving at speeds close to thespeed of light c. We describe this system by a Hamiltonian of the form,

H =

N∑l=1

√p2x + p2y c.

(a) Show that the canonical partition function is

ZN =1

N !

[2πV

(kBT

hc

)2]N

.

(b) Find the Helmholtz free energy A(T, V,N), the entropy S(T, V,N), the pressure p(T, V,N),and the internal energy U(T,N).(c) Find the adiabate (for constant N) and express it in the form pνV = const.(d) Infer from the given canonical partition function ZN (T, V ) an explicit expression for the grandpartition function Z(T, V, µ), where µ = kBT ln z is the chemical potential and z is the fugacity.Use

∫∞0dxxne−ax = n!a−n−1, lnn! ' n lnn− n,

∑∞n=0 x

n/n! = ex.

Solution:

[tex78] Array of classical harmonic oscillators (canonical ensemble)

Consider an array of N 3-dimensional classical harmonic oscillators, representing a system of 3Nuncoupled degrees of freedom:

H =3N∑i=1

(p2

i

2m+

12mω2q2

i

).

(a) Calculate the canonical partition function ZN for this model.(b) Derive from ZN the Helmholtz free energy A(T, N), the entropy S(T, N), the internal energyU(T, N), and the heat capacity C ≡ (∂U/∂T )N .

Solution:

[tex136] Irreversible decompression

Consider an insulating box with two compartments. Each compartment initially contains N atomsof a monatomic classical ideal gas in equilibrium at initial pressures p1 6= p2 and at the same initialtemperature T . Gas atoms are then allowed to leak through a hole in the dividing wall.(a) Show that the temperature remains the same in the final equilibrium state.(b) Find the uniform pressure p in the final equilibrium state as a function of p1 and p2.(c) Find the increase in total entropy, ∆S, between the initial and final equilibrium states.

Solution:

[tex137] Irreversible heat exchange

Consider an insulating box with two compartments. Each compartment initially contains N atomsof a monatomic classical ideal gas in equilibrium at initial temperatures T1 6= T2 and at the sameinitial pressure p. Gas atoms are then allowed to leak through a hole in the dividing wall.(a) Find the uniform temperature T in the final equilibrium state as a function of T1 and T2.(b) Show that the pressure remains the same in the final equilibrium state.(c) Find the increase in total entropy, ∆S, between the initial and final equilibrium states.

Solution:

[tex139] Reversible decompression

Consider a rigid, insulating box with two compartments of volumes V1 and V2 separated by aninternal wall. Each compartment contains N atoms of a monatomic classical ideal gas [pV =NkBT, CV = 3

2NkB ] in equilibrium at the same temperature Tini.(a) Find the maximum work, ∆W (Tini, V1, V2, N), that can be extracted from this system by anymeans that keep the box rigid and insulating.(b) Design a reversible process that employs the internal wall, which is movable by an externalagent in a controlled manner and which can be switched between heat-conducting and insulatingmodes.

Solution:

[tex140] Reversible heat exchange

Consider a rigid, insulating box with two compartments of volumes V1 and V2 separated by aninternal wall. Each compartment contains N atoms of a monatomic classical ideal gas [pV =NkBT, CV = 3

2NkB ] in equilibrium at the same pressure.(a) Find the maximum work, ∆W (T1, T2, N), that can be extracted from this system by any meansthat keep the box rigid and insulating.(b) Design a reversible process that employs the internal wall, which is movable by an externalagent in a controlled manner and which can be switched between heat-conducting and insulatingmodes.

Solution:

[tex141] Heavy piston

A cylinder of cross section A with insulating walls has two compartments separated by a disk ofmass m. The axis of the cylinder is vertical. A uniform gravitational field g is present. The disk isinitially held at a fixed position by an external agent. The upper compartment is evacuated andthe lower compartment contains 1 mol of a monatomic, classical, ideal gas [pV = RT , CV = 3

2R]at temperature T0, volume V0, and pressure p0. When the disk is released, it moves without (wall)friction and comes to rest at a lower position. Calculate the final values p1, V1, T1 of pressure,volume, and temperature, respectively. The disk does not exchange heat. The only significantaction of the gravitational field is on the disk.

Hint: Use energy conservation and Newton’s third law. Assume thermal equilibrium for the initaland final states.

m

p = 0

A

gp0 V

0

T0

Solution:

Ensemble averages [tln52]

All thermodynamic quantities of a closed system can be inferred from thecanonical partition function ZN via the associated thermodynamic potential:

ZN =1

CN

∫Γ

d6NX exp (−βH(X)) , β =1

kBT.

Further properties of the system can be obtained from the canonical proba-bility density ρ(X) via equilibrium expectation values of arbitrary dynamicalvariables f(X):

〈f〉 =

∫Γ

d6NX ρ(X)f(X), ρ(X) =1

ZNCN

exp (−βH(X)) .

From such expectation values, we can recover thermodynamic quantities andcalculate fluctuations thereof, which are related to response functions, i.e.different thermodynamic quantities. Other expectation values, e.g. correla-tion functions, cannot be inferred directly from ZN .

• Uncertainty about microstate and entropy:

S = −kB

∫Γ

d6NX ρ(X) ln[CNρ(X)].

• Average value H and internal energy:

〈H〉 =

∫Γ

d6NX ρ(X)H(X) =1

ZNCN

∫Γ

d6NX H(X) e−βH(X)

⇒ 〈H〉 = − 1

ZN

∂ZN

∂β= − ∂

∂βln ZN =

∂

∂β(βA)

Use∂

∂β=

(∂T

∂β

)∂

∂T= −kBT 2 ∂

∂T⇒ 〈H〉 = A− T

∂A

∂T= A + TS = U.

• Energy fluctuations and heat capacity:

〈H2〉 − 〈H〉2 =1

ZN

∂2ZN

∂β2−

[1

ZN

∂ZN

∂β

]2

=∂

∂β

[1

ZN

∂ZN

∂β

]

⇒ 〈H2〉 − 〈H〉2 =∂2

∂β2ln ZN = −∂U

∂β= kBT 2 ∂U

∂T= kBT 2CV .

Classical virial theorem [tln83]

Classical Hamiltonian system: H = T + V .N interacting particles in 3D space represent 3N degrees of freedom.Phase-space coordinates: xi = (ql, pl), i = 1, . . . , 6N, l = 1, . . . , 3N .

Theorem in general form:⟨xi∂H∂xj

⟩.=

1

Z

∫d6Nx xi

∂H∂xj

e−H/kBT = −kBTZ

∫d6Nx xi

∂e−H/kBT

∂xj.

Integrate by parts:

⇒⟨xi∂H∂xj

⟩=kBT

Z

∫d6Nxe−H/kBT δij = kBTδij.

Equipartition: average kinetic energy per degree of freedom

T =3N∑l=1

p2l2m

⇒⟨pl∂H∂pl

⟩= 〈mp2l 〉 ⇒

⟨1

2mp2l

⟩=

1

2kBT.

Virial: pair interactions

V =1

2

∑l 6=l′

v(|ql − ql′|). Set qll′.= ql − ql′ .

⇒ 1

6

∑l 6=l′

⟨qll′

∂v

∂qll′

⟩= NkBT − pV.

Anharmonic crystal in 1D: average potential energy per bond

V =N−1∑l=1

1

2u|ql − ql+1|ν with ν > 0. Set p = 0.

⇒⟨1

2u|ql − ql+1|ν

⟩=kBT

ν.

[adapted from Schwabl 2006]

Systems of noninteracting particles [tln54]

Consider a classical systems of N noninteracting particles.

Hamiltonian: H =N∑

l=1

hl(ql,pl).

Canonical partition function of distinguishable particles:

ZN =1

CN

∫Γ

d6NX e−βH(X) =N∏

l=1

Zl, Zl =1

h3

∫d3ql d

3pl e−βhl(ql,pl).

Factorizing phase-space probability density:

ρ(X) =1

ZNCN

e−βH(X) =N∏

l=1

[1

h3Zl

e−βhl(ql,pl)

].

Identical one-particle Hamiltonians:

h1 = · · · = hn ≡ h(q,p) ⇒ Z1 = · · · = ZN = Z ≡ 1

h3

∫d3q d3p e−βh(q,p).

⇒ ZN = ZN , ρ(X) =N∏

l=1

[1

h3Ze−βh(ql,pl)

].

Indistinguishable particles:

ZN =1

N !ZN .

Note: It is important that we discriminate between noninteracting subsys-tems that are identical but distinguishable (e.g. atoms vibrating about rigidlattice sites) and noninteracting subsystems that are identical and indistin-guishable (e.g. atoms of an ideal gas).

Further ensemble averages [tln55]

Probability density in one-particle phase space:

ρl(q,p) = 〈δ(ql − q)δ(pl − p)〉.

Position distribution and momentum distribution:

ρl(q) = 〈δ(ql − q)〉, ρl(p) = 〈δ(pl − p)〉.

Distribution of distances and relative momenta between pairs of particles:

flm(r) = 〈δ(r − |ql − qm|)〉, Flm(P ) = 〈δ(P − |pl − pm|)〉.

Average distance between pairs of particles:

〈rlm〉 = 〈|ql − qm|〉 =

∫ ∞

0

dr rflm(r).

Average magnitude of relative momentum between pairs of particles:

〈Plm〉 = 〈|pl − pm|〉 =

∫ ∞

0

dP PFlm(P ).

Applications to the classical ideal gas:

Noninteracting particles: h(q,p) =p2

2m⇒ Z =

V

λ3T

, λT =

√h2

2πmkBT.

⇒ ρl(q,p) = V −1(2πmkBT )−3/2e−p2/2mkBT .

⇒ ρl(q) =

∫d3p ρl(q,p) =

1

V.

⇒ ρl(p) =

∫d3q ρl(q,p) = (2πmkBT )−3/2e−p2/2mkBT .

ρl(p)d3p = f(v)d3v ⇒ f(v) =

(m

2πkBT

)−3/2

e−mv2/2kBT .

The spatial distribution of ideal gas particles in a uniform gravitational field(law of atmospheres) is calculated in exercise [tex79].

The distribution of relative momenta between pairs of ideal gas particles iscalculated in exercise [tex80].

[tex79] Classical ideal gas in uniform gravitational field

Consider a column with cross-sectional area A of a classical ideal gas (N atoms of mass m) in auniform gravitational field of magnitude g. The gas is in thermal equilibrium at temperature T .The Hamiltonian reads:

H =N∑

l=1

(p2

l

2m+mgzl

),

where zl is the height of particle l above sea level.(a) Find the probability density ρ1(z) for the vertical positions of individual gas atoms.(b) Find the pressure distribution p(z).

Solution:

[tex135] Gas pressure and density inside centrifuge

Consider a hollow disk of width L and radius R filled with N particles of a dilute gas at temperatureT . The disk is in a state of rotation with angular velocity ω about its axis.(a) Find the probability density ρ1(r) for the radial position of a gas particle and find the particledensity n(r). Note that the unit of ρ1(r) is [m−2] and the unit of n(r) is [m−3].(b) Find the pressure p(r).(c) In an experiment that measures p(0) and p(R) at various values of ω and fixed T , which twoquantities must be plotted against each other such that the data points are predicted to fall ontoa straight line with slope equal to the mass of the gas particles?

Solution:

[tex80] Relative momentum of two ideal gas particles

Consider a classical ideal gas of N atoms with mass m confined to a box of volume V in thermalequilibrium with a heat reservoir at temperature T .(a) Find the distribution Flm(P ) ≡ 〈δ(P − |pl − pm|)〉 of the magnitude of the relative momentaof two ideal gas particles.(b) Find the ratio of the average magnitudes 〈P 〉/〈p〉 of the relative momentum of two particlesand the momentum of a single particle.

Solution:

Partition function and density of states [tln56]

Why do the microcanonical and canonical ensembles yield the same results?

(a) Derivation of ZN from Ω(U, V,N).

Relation between the microcanonical phase-space volume Ω(U, V,N) and thenumber of microstates Σ(U, V,N) up to the energy U :

Ω(U, V,N) ≡∫

H(X)<U

d6NX = CNΣ(U, V,N).

Density of microstates:

g(U) =∂Σ

∂U.

The canonical partition function is then obtained via Laplace transform:∫ ∞

0

dU g(U)e−βU =1

CN

∫Γ

d6NX e−βH(X) = ZN .

Here the energy scale has been shifted such that U0 = 0.

(b) Derivation of Ω(U, V,N) from ZN .

Complex continuation of the canonical partition function:

ZN = Z(β) for β = β′ + iβ′′ with β′ > 0.

The microcanonical phase-space volume is the obtained via inverse Laplacetransform:

g(U) =1

2πi

∫ β′+i∞

β′−i∞dβ eβUZ(β), Ω(U, V,N) = CN

∫ U

0

dU ′ g(U ′).

Both calculations are carried out in exercise [tex81] for the classical ideal gas.

[tex81] Ideal gas partition function and density of states

(a) Starting from the result of [tex73] for the phase-space volume Ω(U, V,N) of a classical ideal gas(N particles with mass m) in the microcanonical ensemble, calculate the density of microstates,gN (u), and then, via Laplace transform, the result of [tex76] for the canonical partition functionZN (β), where β = 1/kBT .(b) Starting from the canonical partition function ZN (β) analytically continued into the complexplane, calculate the density of state gN (U) via inverse Laplace transform.

Solution:

Vibrational heat capacities of solids [tln57]

The interaction between atoms is attractive at long distances and repulsiveat short distances. The lowest-energy configuration of a macroscopic systemof N atoms is a perfect lattice. This is the equilibrium state at T = 0. Ithas zero entropy. Heat input δQ = CdT causes lattice vibrations. In thefollowing we study vibrational heat capacities in successively improved ap-proximations.

Atoms bound to rigid lattice by harmonic force (classical model):

The theory of Dulong and Petit considers an array of N classical 3D harmonicoscillators with identical angular frequencies. The resulting vibrational heatcapacity, C = 3NkB, is T -independent and is calculated in exercise [tex74] fora microcanonical ensemble and in exercise [tex78] for a canonical ensemble.

The main insufficiency of the Dulong-Petit result is that C does not approachzero in the low-temperature limit, in violation of the third law.

Atoms bound to rigid lattice by harmonic force (quantum model):

The theory of Einstein considers an array of N quantum 3D harmonic os-cillators with identical angular frequencies ω. The resulting vibrational heatcapacity,

C =

(ΘE

kBT

)23NkBeΘE/T

(eΘE/T − 1)2 , kBΘE = ~ω,

goes to zero exponentially in the low-T limit, C ∼ e−ΘE/T , and approachesthe Dulong-Petit result, C = 3NkB, at high T . Einstein’s result is derivedin exercise [tex75] for the microcanonical ensemble and in exercise [tex82] forthe canonical ensemble.

The main insufficiency of Einstein’s result is that it contradicts experimentalevidence, which suggests C ∼ T 3 at low T .

1

Atoms interacting via harmonic force:

H =3N∑i=1

p2i

2m+

∑ij

Aijqiqj =3N∑i=1

[p2

i

2m+

1

2mω2

i Q2i

].

Here Aij is the dynamical matrix. The second equation results from atransformation to normal-mode coordinates. In the present context the nor-mal modes are sound waves (phonons).

Quantum mechanically, this system is an array of 3N independent harmonicoscillators with normal mode frequencies ωi:

H =3N∑i=1

~ωi

(ni +

1

2

), ni = 0, 1, 2, . . .

The resulting Helmholtz free energy (in generalization to the result derivedin [tex82]) reads:

A =1

2

3N∑i=1

~ωi + kBT3N∑i=1

ln(1− e−β~ωi

).

In Debye’s theory, the normal modes, which, in general, consist of multiplebranches of acoustic and optical phonons, are replaced by a single branch ofsound waves with linear dispersion ω = ck as is expected in a continuousisotropic elastic medium.

Total number of modes: 3N (same as in original lattice model).

Density of modes in k-space: V/(2π)3.

Number of polarizations: 3 (2 transverse, 1 longitudinal).

Number of modes in dω: n(ω)dω =V

8π3(3)(4π)

ω2

c2

dω

c=

3V

2π2c3ω2dω.

Debye frequency:3V

2π2c3

∫ ωD

0

dω ω2 = 3N ⇒ ω3D =

6Nπ2c3

V.

Density of modes: n(ω) =9N

ω3D

ω2.

The resulting vibrational heat capacity is calculated in exercise [tex83] anddoes show the experimentally observed ∼ T 3 behavior as T → 0:

C = 9NkB

(T

ΘD

)3 ∫ ΘD/T

0

dxx4ex

(ex − 1)2 , ΘD = ~ωD/kB.

2

[tex82] Array of quantum harmonic oscillators (canonical ensemble)

Consider an array of N 3-dimensional quantum harmonic oscillators:

H =3N∑i=1

[~ω

(ni +

12

)], ni = 0, 1, 2, . . . .

(a) Calculate the canonical partition function ZN for this model.(b) Derive from ZN the Helmholtz free energy A(T, N), the internal energy U(T, N), and the heatcapacity C = (∂U/∂T )N .(c) Show that U(T, N) approaches the result of [tex78] for the classical oscillators.(d) Calculate the quantity 〈ni〉 for a single degree of freedom. It reflects the average number ofelementary energy quanta that are excited in one oscillator when it is in thermal equilibrium attemperature T .

Solution:

Vibrational heat capacities of solids [tsl29]

Density of vibrational modes in aluminum

Debye predicition for heat capacityin comparison with experimental data

[from Garrod 1995]

Thermodynamic perturbation expansion [tln80]

Consider a classical dynamical system in the canonical ensemble.

H = H0 + V.

The term H0 represents the dominant contribution to the energy of the sys-tem under the circumstances of interest. We assume that the Helmholtz freeenergy, A0(T, V, N), for that part alone can be calculated exactly:

e−βA0 =

∫dΓ e−βH0 , dΓ

.= d6NX, β

.=

1

kBT.

We can then treat V perturbatively via the following expansion:

e−βA =

∫dΓ e−β(H0+V ) '

∫dΓ e−βH0

(1− βV +

1

2β2V 2

).

This expression is then further expanded by using ln(1− x) ' −x + x2/2:

−βA ' ln

(e−βA0 − β

∫dΓ V e−βA0 +

1

2β2

∫dΓ V 2e−βH0

)' −βA0 + ln

(1− β

∫dΓ V eβ(A0−H0) +

1

2β2

∫dΓ V 2eβ(A0−H0)

).

⇒ A = A0 +

∫dΓ

(V − 1

2βV 2

)eβ(A0−H0) +

1

2β

[∫dΓ V eβ(A0−H0)

]2

.

With ensemble averages,

〈V 〉 .=

∫dΓ V e−βH0∫dΓ e−βH0

=

∫dΓ V eβ(A0−H0), 〈V 2〉 =

∫dΓ V 2eβ(A0−H0),

and the relation〈V 2〉 − 〈V 〉2 = 〈(V − 〈V 〉)2〉,

we can write

A = A0 + 〈V 〉 − 1

2β〈(V − 〈V 〉)2〉.

The criterion of applicability for this expansion is 〈V 〉/N kBT . Note thatif 〈V 〉 = 0 then the leading-order perturbation always reduces the Helmholtzfree energy.

[tex83] Vibrational heat capacity of a solid

The vibrational Helmholtz free energy of a harmonic crystal of N atoms in thermal equilibrium attemperature T is

A =12

3N∑i=1

~ωi + kBT

3N∑i=1

ln(1− e−β~ωi

),

where the ωi are the normal modes of transverse and longitudinal lattice vibrations (phonons). InDebye’s theory, the density of modes is approximated by the functions n(ω) = 9Nω2/ω3

D, wherethe Debye frequency ωD is an undetermined parameter.(a) Show that the internal energy U = A+ TS in the Debye approximation reads

U =98N~ωD + 9NkBT

(T

ΘD

)3 ∫ ΘD/T

0

dxx3

ex − 1,

where ΘD = ~ωD/kB is called the Debye temperature.(b) Derive an expression for the heat capacity C = (∂U/∂T )N .(c) At low temperatures the upper boundary ΘD/T in the above integral may be replaced byinfinity,

∫∞0dxx3/(ex − 1) = π4/15. Use this fact to determine the leading low-temperature term

of the heat capacity C.

Solution:

[tex104] Anharmonic oscillator and thermodynamic perturbation

Consider an array of N one-dimensional anharmonic oscillators,

H =N∑

l=1

[p2

l

2m+ V (ql)

], V (q) = cq2 − gq3 + fq4.

(a) Evaluate the canonical partition function perturbatively by treating the quadratic term of Vexactly and considering only the leading nonzero corrections of the cubic and the quartic terms.(b) Show that the heat capacity in this approximation is C/NkB = 1 + [15g2/16c3 − 3f/4c2]kBT .(c) Show that the mean displacement in this approximation is 〈q〉 = (3g/4c2)kBT .

Solution: